Official Article on Solar Intervals

7/27/2019 Official Article on Solar Intervals

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Carlos Barrera Atuesta, 2004-2013. All Rights Reserved.

SOLAR INTERVALS OF THE DRESDEN CODEX VENUS TABLECARLOS BARRERA ATUESTA, Independent Researcher on Maya Science

Introduction

Could the Maya of the Classic Period (from third to tenth centuries) predict astronomicalevents hundreds and even thousands of years in advance? How did they manage to obtainthe precision that is implied in their astronomical calculations? These are mysteries thatpersist after more than one century of research. The recent discovery of a solar propertyencoded in a set of multiples of the Dresden Codex - the oldest manuscript from theAmerican continent would reveal key aspects for understanding their mathematicalmodels and scientific foundations, partially rescue their intangible heritage, and confirm thevalidity of the GMT correlation between Maya and Christian dates. These results suggestthat the same calculation principle used to accurately determine, throughout the centuries,the main solar references of the agricultural year, also would have been used to set suitable

dates for political, social, ritual, and mythical events, dates of deep time and the distantfuture.

Abstract

The ancient Maya are widely recognized for their refined intellectual achievements inAstronomical Sciences1 and Mathematics,2 and it has even been said that their solarcalculations were more accurate than the current Gregorian calendar.3 While it is true thatthere is evidence supporting the foregoing assertions, is also equally true that themathematical procedures or theoretical foundations that allowed them to reach a certaindegree of accuracy are still unknown. In that sense, this article seeks to contribute newconcepts in regard to the Maya conception of time4, extracted from their own epigraphic

records, which could give us a deeper understanding of their scientific methods, and wouldalso help solve other crucial aspects in the study of their Culture, related to Chronology,History and Mythology.

1. On the Calculation of Solar Years

The first indications that suggest the precision of the Maya solar calendar come from worksof John E. Teeple in which he investigates 6940-day intervals between Maya chronologicalrecords that would represent 235 lunations and 19 solar years.5 Teeple also considers thepossibility that the Maya may have allowed the independent course of the Jaab calendarand tropical year through a full synchronization circuit of 29 18980-day Calendar Rounds,6where the following identity is obtained: 1508 x 365d = 1507 x 365.2422d.

Regarding the techniques used by the ancient Maya to verify the moments in which themain solar references of the agricultural year were observed, the alignments of stelaeoriented towards specific points in the eastern and western horizon, as well as the design ofarchitectural structures arranged to indicate the dates of occurrence of solstices and theequinoxes7 can be cited, as evidenced by Group E at Uaxactn (Figure 1), the Temple ofthe Seven Dolls of Dzibilchaltn, El Castillo at Chichn Itz, and El Castillo at Mayapn.


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FIG. 1. Architectural layout of the structures that integrate the E Group at Uaxactun, with theirrespective description of solar dates and references [Drawing by Robert Sharer]8

Among the technical resources used by the ancient Maya to find out the fleeting momentwhen the Sun was at its highest point in the sky, were the vertical zenith sighting tubes:dark underground chambers that project a perpendicular beam of light towards the center ofits base when the rays of the Sun pass through an observation tube in the roof of thestructure; a phenomenon that occurs only twice a year in regions between the Tropic ofCancer and the Tropic of Capricorn, and that is called "the zenith passage of the Sun".

Archaeological evidence on the tracking of the zenith passage of the Sun in

Mesoamerica has been found in the so-called Cave Los Amates at Xochicalco, and thezenithal observatories of Monte Alban and Teotihuacan.9Pioneering studies conducted by Zelia Nuttal,10 Ola Apenes11 and Vincent H.

Malmstrm,12 linked the zenith passages of the Sun that occurred on Mesoamerican regionslocated between 1442'N and 15N with a possible astronomical origin of theMesoamerican calendar. According to the above criteria, the zenith passages of the Sunwould occur in these latitudes at noon on August 12-13, and April 30-May 1, there being agap between these events of 260 days, and later, of 105 days. The agricultural importanceof the dates obtained lies in the fact that they mark out seasons of rain and drought inextensive regions of Mesoamerica, associated with the planting and harvesting of corn.

The establishment of the above solar references, solstices and equinoxes included,

allows for a better understanding of the basic mathematical characteristics that define theidealized 364-day solar model, which the Maya would have formulated based on thirteen-day periods (Figure 2). The midpoint between April to August solar zenith passagesrepresents the summer solstice whereas the midpoint between August to April zenith solarpassages represents the winter solstice. In turn, the midpoint between summer to wintersolstices represents the autumnal equinox, while the midpoint between winter to summersolstices represents the vernal equinox.


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FIG. 2. 364-day solar representation model. References nearest to the equinoxes are 39 daysaway, and those nearest to the solstices are 52 days away.

When making the respective calculations it is observed that the separation between thezenith passages of the Sun and the summer solstice can be defined by four thirteen-dayperiods, and the separation between equinoxes and their nearest zenith passages can bedefined by three thirteen-day periods. In order to obtain a symmetric model of solarrepresentation, the positions located 52 days from the winter solstice are also considered,

and 39 days from their immediate equinoxes, representing these references, the passages ofthe Sun through the nadir,13 which take place at midnight on October 29-31, and onFebruary 9-11.

In actual astronomical terms, the existing separation for the Northern Hemispherebetween the equinoxes and the winter solstice is 89-90 days, and between the equinoxesand the summer solstice, of 93-94 days, the 21st days of the respective months of June andDecember being the usual dates for the solstices, whereas the vernal equinox tends to occurmore towards March 20, and the autumnal equinox near September 22.

This interesting asymmetry of the solar year would have been resolved by the ancientMaya based on the Mesoamerican 260-day calendar, as evidenced by a study of modulararithmetic applied to the so called anomalous multiples of the Venus Table. Page 24 of the

Dresden Codex (Figure 3) presents a general synthesis of this structure.14

2. On the Structure of the Venus Table

The recording of eight 360-day Tuns, two 20-day Winalsand zero Kins or days (8.2.0), inthe lower right section of page 24, is equivalent in our decimal system to 2920 days: aninterval that represents the synchronization of eight 365-day Jaab calendars, and fivecanonical 584-day Venus cycles.15 When such transition is made from a 1-Ajaw origindate, it leads to a 9-Ajaw destination date. The following records to its left correspond to


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the second, third and fourth multiples, and the four upper entries represent the fifth toeighth multiples. Going up through the Table, we have the ninth to twelfth multiples, thenintroducing a discontinuity between the twelfth (4.17.6.0) and thirteenth (5.5.8.0) 2920-daymultiples, caused by a set of records in between, referred to as anomalous multiples.

FIG. 3. Mathematical decoding of page 24 of the Dresden Codex. The set of anomalous multiplesof 9100, 33280, 68900 and 185120 days, had traditionally been interpreted exclusivelyaccording to Venus cycles. [Photo by Justin Kerr for FAMSI]


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The anomalous multiples (or peculiar numbers) of the Venus Table, in contrast to the2920-day multiples, cannot be expressed as whole repetitions of the 365-day Jaab calendar, or the canonical 584-day Venus cycle, but they do allow for their exactdivisibility by the Mesoamerican 260-day calendar, which is their main characteristic. The

intervals represented by anomalous multiples are: 9100 (1.5.5.0), 33280 (4.12.8.0), 68900(9.11.7.0) and 185120 days (1.5.14.4.0).The upper right section of the Table shows the first to fourth 37960-day multiples

(5.5.8.0 to 1.1.1.14.0), representing the moments of synchronization between theMesoamerican calendar, the Jaab' calendar and the canonical cycle of Venus, being 37960days (i.e. 13 x 2920d) the least common multiple of 260, 365 and 584 days. Anothersignificant interval included within the structure, is that of 26280 days (i.e. 9 x 2920d =3.13.0.0), which represents the synchronization between 73 360-day Tun periods, 72 365-day Jaab' calendars, and 45 584-day Venus cycles.

The establishment of the historical moment referred to in the above records can bedetermined by adopting a correlation constant between Maya and Christian dates,16 and byperforming the operations indicated in the lower left section of page 24, by Ring Number6.2.0, and the Long Round 9.9.16.0.0 to its right. The subtraction of the 2200-day RingNumber, from the Era Date 0.0.0.0.0, 4 Ajaw 8 Kumk'u, and the subsequent application ofthe 1366560 days implied by the aforesaid Long Round, leads towards the main record ofthe Venus Table, defined by the date 9.9.9.16.0, 1 Ajaw 18 K'ayab.17

00.00.00.00.00 4 Ajaw 8 Kumku (Era Date)00.00.06.02.00 2200-day Ring-Number

12.19.13.16.00 1 Ajaw 18 Kayab (Pre-Era Date)+09.09.16.00.00 1366560-day Long Round09.09.09.16.00 1 Ajaw 18 Kayab (Main Record)

3. On the Calculation of the Venus Cycle

Venus actual synodic period is 583.921375 days, hence, there is a cumulative error due toexcess of about 5.11 days in the 37960-day interval, since 37960d mod 583.921375d =5.11062d. For this reason, John E. Teeple suggested that the peculiar multiples of the VenusTable could play a compensatory role on such deviation.18

Teeple noticed that by subtracting 33280 days from 68900 days, a slightly higher valuewas obtained than the one generated by 61 actual Venus periods, and that by accumulatingfour 35620-day repetitions, it was possible to include an additional 33280-day interval,forming a precision 175760-day transition, equivalent to 301 583.9203-day Venus cycles.

(68900d

33280d

) = 35620 days = 61 583.921375-day Venus periods + 0.796125d

(4 x 35620d + 33280d) = 175760 days = 301 583.921375-day Venus periods0.3d175760 days = 301 583.9203-day Venus periods (on average)

In 2007, the author observed that this same 175760-day transition could be obtained byalternating (33280d + 37960d + 33280d + 37960d + 33280d), or (68900d + 37960d + 68900d).By doing so, Venus position could be indefinitely projected in time and would never showa theoretical deviation greater than 3.52 days or 2.72 days, relative to its point of origin.


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The last day in which Venus is visible at sunset, and its first day of visibility at the breakof dawn, typically occur within 4 days of their inferior conjunction, which makes possibleits tracking through the above-described method.19 The mathematical basis behind Venusbehavior was established by Lounsbury,20 by proving that all peculiar numbers can beobtained from a linear Diophantine equation of the type z = 37960x2340y.

First peculiar number: 1.5.5.0 = 9100 days = (4 x 37960d61 x 2340d)Second peculiar number: 4.12.8.0 = 33280 days = (1 x 37960d2 x 2340d)Third peculiar number: 9.11.7.0 = 68900 days = (2 x 37960d3 x 2340d)Fourth peculiar number: 1.5.14.4.0 = 185120 days = (5 x 37960d2 x 2340d)

Eric Thompson believed that the anomalous 9100-day multiple was in error and shouldbe replaced with an interval of 4 x 2340d = 9360d.21 Without knowing it, Thompsonjustified this replacement using the same mathematical logic that would years later confirmthe validity of the anomalous multiple that it sought to correct.22

Comparing Teeple's 175760-day interval to the anomalous 185120-day multiple,Thompson realized that the 9360 days between them, projected the position of Venus in9.9.9.16.0, towards a future heliacal rise,23 as indicated by the correlation between Mayaand Gregorian dates that Thompson himself and John Teeple had helped improve in 1927,based on the works of Joseph Goodman and Juan Martinez in 1897.24

The solution trajectory proposed by Lounsbury for the Venus Table used, in fact, the185120-day peculiar number, and the third 37960-day multiple (i.e. 113880 days),25 bothrecorded on page 24 of the Dresden Codex, to obtain a compensating effect similar to thatgenerated by the 9360-day interval on Venus synodic position of 9.9.9.16.0.

185120 days = 317 583.921375-day Venus cycles + 16.924125d113880 days = 195 583.921375-day Venus cycles + 15.331875d9360 days = 16 583.921375-day Venus cycles + 17.258d

The validity of the anomalous 9100-day multiple had been suggested by Bryan Wells in1991, who stated two properties that made it special: [1] its exact divisibility by the Maya1820-day cycle, equivalent to 7 x 260d and 5 x 364d, and [2] the remainder of 340 daysobtained when the 9100-day interval is compared to 15 canonical 584-day Venus cycles.26

On pages 46-50 of the Dresden Codex, the duration of the Venus cycle is defined fromfour sub-intervals of 236, 90, 250 and 8 days. The obtained remainder of 340 days couldthen be expressed as (90d + 250d).

9100 days = 15 x (90d + 250d + 8d + 236d) + (90d + 250d) = 15 x 584d + 340d

The way in which the anomalous 9100-day multiple ought to be incorporated into theVenus Table was established by the author in 2007 by noticing that there were only two 1-Ajaw dates in its structure, separated by 9100 days (represented by registers 165 and 227 of260), and that the achieved target date (register 227) should correspond to the last day onwhich Venus was visible as evening staron 9.10.2.16.0, 1 Ajaw 13 Kankin.27

9.8.17.11.0, 1 Ajaw 18 Muwan + 9100d= 9.10.2.16.0, 1 Ajaw 13 Kankin


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A subsequent calendric operation showed that, when making a 9100-day transition from9.9.9.16.0, an important computing reference was obtained, associated with a South-yellowquadrant of the Maya Cosmos: the 819-day station 9.10.15.3.0, 1 Ajaw 13 Pax.

9.9.9.16.0, 1 Ajaw 18 Kayab + 9100

d

= 9.10.15.3.0, 1 Ajaw 13 PaxFor some reason that should be investigated, there is a separation of 42 x 32760 days

(3767 sidereal years) between 9.10.15.3.0, 1 Ajaw 13 Pax, and a mythical 819-day station(12.19.13.3.0, 1 Ajaw 18 Sotz) recorded in Palenque during the Classic Period.28

4. The Discovery of the Solar Intervals of the Venus Table

The expression 68900d mod 365.2422d = 234.4664d represents the actual lapse generated bythe third anomalous multiple of the Venus Table on the solar cycle. The operation indicatesthat after 68900 days, 188 365.2422-day solar years are completed, leaving a remainder of234.4664 days, which is equivalent to the actual time elapsed since the beginning of the

solar cycle. Another way to interpret the same result is to state that there are 130.7758 daysleft = (365.2422d234.4664d), to complete 189 tropical years and recover the original date.Within the idealized 364-day solar model (see Figure 2), the obtained lapse of 234 days

represents a transition between opposite dates (separated by 182 days), plus 52 days,whereas going back 130 days = (364d 234d), represents a transition between oppositedates minus 52 days. From the foregoing, it can be concluded that the anomalous 68900-day multiple makes it possible to link the April and August zenith passages of the Sun withthe winter solstice on December 21, as well as the October and February solar nadirs withthe summer solstice29 on June 21.

Applying the 584285 correlation constant, according to which the Initial Series11.16.0.0.0, 13 Ajaw 8 Xul is equivalent to the Gregorian date of A.D. 1539 November

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the main record of the Venus Table 9.9.9.16.0, 1 Ajaw 18 K'ayab, corresponds to thesolar nadir on A.D. 623 February 9.31Consequently, subtracting the anomalous 68900-day multiple from 9.9.9.16.0, should

result in the summer solstice, and subtracting 68900 days therefrom, should result in theOctober solar nadir. Whenever the reached date differs from the expected value in one day,it could be corrected by applying the canonical 364-day cycle in the correspondingdirection.

The 1364360-day interval, applied to the Era Date 0.0.0.0.0 to obtain record 9.9.9.16.0could be interpreted, in fact, as a huge transition of 3735 years, between the zenithpassage of 3114 B.C. August 13, and the solar nadir on A.D. 623 February 9.

9.9.9.16.00.0.0.0.0 = 1364360 days Potential solar transition1364360 days = (1366560d2200d) Maya notation 9.9.9.16.01364360d mod 365.2422d = 180.38d Effective progress in daysAugust 13 (zenith passage) + 180.38d February 9 (solar nadir)

Similar analyses show that the peculiar 33280-day interval could also be used to performtransitions from the vernal equinox to the May 1 zenith passage, or from the August zenithpassage to the September 23 autumnal equinox.


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Concatenating the anomalous multiples of 33280 and 68900 days, a transition isobtained from the vernal equinox to the winter solstice with a stopover on the April 30-May1 zenith passage, and when concatenating them in the reverse order, a transition is obtainedfrom the winter solstice to the autumnal equinox with a stopover on the August 12-13zenith passage.

Vernal equinox + 33280d = April-May zenith passageApril-May zenith passage + 68900d = Winter solstice

Winter solstice + 68900d = August zenith passageAugust zenith passage + 33280d = Autumnal equinox

A new analysis of modular arithmetic, this time applied to the concatenated anomalousmultiples of (185120d + 9100d), reveals that this interval could also be used to perform thesame transitions from equinoxes to solstices obtained through the anomalous multiples of(33280d + 68900d), there being a difference between their respective remaining results ofjust one day. As a consequence, said intervals may be conveniently combined, duplicated orsubtracted, depending on the specific situation at hand.

(185120d + 9100d) mod 365.2422d =88.8504d(33280d + 68900d) mod 365.2422d =87.8160d

Typical separation between the vernal equinox and the winter solstice:89 daysTypical separation between the winter solstice and the autumnal equinox:89 days

The huge 1366560-day interval, which finally leads to the main entry of the Venus Table9.9.9.16.0, actually serves the same solar function obtained by concatenating all theanomalous multiples, since 1366560d mod 365.2422d = 188.9298d, whereas (33280d +68900d + 185120d + 9100d) mod 365.2422d = 188.5758d. Therefore, the 9.9.16.0.0 intervalof 1366560 days could be used to perform transitions of 3741 solar years from acomputing vernal equinox (March 18/19) to an autumnal equinox (September 22/23).

March 18/19 + 33280d = April-May zenith passageApril-May zenith passage + 68900d = Winter solsticeWinter solstice + (185120d + 9100d) = September 22/23March 18/19 + 1366560d (9.9.16.0.0) = September 22/23

Considering the Calendar Round 1 Ajaw 18 Wo, adjacent to 1 Ajaw 18 K'ayab on page24 of the Dresden Codex, it can be observed that the minimum distance between them is4680 days: a renowned Maya interval equivalent to 2 x 2340d, 13 x 360d, 18 x 260d, andother equivalents of an astronomical nature. As a result, by subtracting 4680 days from9.9.9.16.0, the Initial Series 9.8.16.16.0, 1 Ajaw 18 Wo is obtained.

Maya date 9.8.16.16.0 is correlated to the Gregorian date of A.D. 610 April 18, which ismeaningful, because Venus is located at its first stationary position (the point at whichVenus begins its 40-day retrograde motion). A.D. 610 April 18 is also the 18980-dayCalendar Round anniversary of the zenith passage of the sun on A.D. 558 April 30, whichtook place during the 819-day station 9.6.4.3.0, 1 Ajaw 18 Wo.


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Likewise, A.D. 610 April 18 precedes A.D. 635 March 18 by 9100 days, and this date inturn, precedes the zenith passage of A.D. 726 April 30 by 33280 days, which suggests anorderly sequence of applied anomalous multiples, through various solar stations, from A.D.610 April 18 (1 Ajaw 18 Wo) to A.D. 1446 September 22 (autumnal equinox).

9.8.16.16.0, 1 Ajaw 18 Wo = A.D. 610 April 18 (Dresden Codex, p.24)A.D. 610 April 18 + 9100d = A.D. 635 March 18 (Computing equinox)A.D. 635 March 18 + 33280d = A.D. 726 April 30 (Solar zenith passage)A.D. 726 April 30 + 68900d = A.D. 914 December 20 (Winter solstice)A.D. 914 December 20 + (185120d + 9100d) = A.D. 1446 September 22 (Equinox)

Finally, when a 2340-day interval is applied to A.D. 635 March 18, the zenith passage ofA.D. 641 August 13 is obtained (which best explains the concept ofcomputing equinox).

2340d mod 365.2422d = 148.5468d33280d mod 365.2422d = 42.9598d

Distance between March 20 equinox and August 13 zenith passage: 146 daysDistance between March 20 equinox and April 30 zenith passage: 41 days

Optimal computing point for the 2340 and 33280-day intervals: March 18Other possible application points (as subtractions): August 15 / May 2

In fact, September 25 may also be considered as a computing autumnal equinoxbecause the operation (September 25 2340d) leads to an April zenith passage, while(September 2533280d) leads to an August zenith passage.

Distance between April 30 zenith passage and September 23 equinox: 146 daysDistance between August 13 zenith passage and September 23 equinox: 41 day

Optimal computing point for the 2340 and 33280-day intervals: September 25Other possible application points (as additions): August 11 / April 28

In 2007, the author deducted an original Venus structure, beginning on the Initial Series9.5.10.8.0, 1 Ajaw 8 Sak, and ending 37960 days after, on 9.10.15.16.0, 1 Ajaw 8 Sak.According to the 584285 correlation constant, the completion of this structure correspondsto the Gregorian date of A.D. 648 September 25, and its origin, to A.D. 544 October 20. 32

Within this framework of ideas, 33280 days before 9.10.15.16.0, Venus should havebeen located three days after its first day of visibility as morning star, during the Augustzenith passage of A.D. 557 (which is also located 4680 days after A.D. 544 October 20).

33280d mod 583.921375d =3.52d9.10.15.16.0, 1 Ajaw 8 Sak: first day on which Venus is visible as morning star9.10.15.16.0, 1 Ajaw 8 Sak33280d = 9.6.3.8.0, 1 Ajaw 3 Mol (A.D. 557 August 13)9.6.3.8.0, 1 Ajaw 3 Mol: first day on which Venus is visible as morning star + 3 days


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Therefore, one Mesoamerican 260-day calendar after 9.6.3.8.0, 1 Ajaw 3 Mol, Venusshould have been very close to its last day of visibility as morning star, during the Aprilzenith passage of A.D. 558, which took place on the aforementioned 819-day station9.6.4.3.0, 1 Ajaw 18 Wo.

9.6.3.8.0, 1 Ajaw 3 Mol + 260

d

= 9.6.4.3.0, 1 Ajaw 18 Wo (A.D. 558 April 30)9.6.4.3.0: First day of Venus as morning star + 263 days / solar zenith passageDistance between the first and last day on which Venus is morning star: 263 days9.6.4.3.0: Last day on which Venus is visible as morning star / solar zenith passage

Now, this Venus-Solar Structure of the Dresden Codex makes complete sense, because2340 days before 9.10.15.16.0, Venus should have been in inferior conjunction with theSun, during the April zenith passage of A.D. 642; (33280d 260d) before 9.10.15.16.0,Venus should have been on its last day of visibility as morning star, during the April zenithpassage of A.D. 558; and (9100d + 250 d + 8d) after 9.9.9.16.0, Venus should have been onits first day of visibility as morning star, during the autumnal equinox of A.D. 648.

9.9.9.16.0, 1 Ajaw 18 Kayab + 9100 = 9.10.15.3.0, 1 Ajaw 13 Pax9.10.15.3.0, 1 Ajaw 13 Pax: Completion of a 90-day Venus sub-cycle9.10.15.3.0, 1 Ajaw 13 Pax: 819-day station and 32760-day anniversary9.10.15.3.0, 1 Ajaw 13 Pax + (250d + 8d) = 9.10.15.15.18, 12 Etznab 6 Sak9.10.15.15.18, 12 Etznab 6 Sak: Autumnal equinox of A.D. 648 September 239.10.15.15.18, 12 Etznab 6 Sak: Completion of an 8-day Venus sub-cycle / stationCompletion of an 8-day Venus sub-cycle: first day on which Venus is the morning star

Selected Results and Conclusions

The implications derived of the above analyses, though diverse, can be summarized as

follows:

[1] The tracking of solar years in Maya chronology is inferred through the interval of(18980d + 9100d + 33280d) = 168 years (which is the distance between the April zenithpassages of A.D. 558 and A.D. 726), the interval of (33280d260d2340d) = 84 years (i.e.the distance between the April zenith passages of A.D. 558 and A.D. 642), and through thepreviously suggested operation (185120d + 9100d)(33280d + 68900d) = 252 years.

[2] Other non-anomalous 260-day multiples, such as 4680 days and 18980 days, as well asother unconventional intervals are also part of the solar calculations. It is especially worthnoting the interval of (26280d + 33280d + 37960d), equivalent to 267 years, and Venus sub-

intervals of 236, 90, 250, and 8 days, which comprise the following solution sequence:

[3] 10.5.6.4.0, 1 Ajaw 18 Kayab (heliacal rise of Venus on 934 November 25)3337960d(830 December 20; inferior conjunction of Venus) 33280d (heliacal rise of Venus;anniversary of 118 x 260d + 29 x 37960d, of 1.18.5.4.0)3426280d (transit of Venus351d,on 667 November 25; new moon) 250d (667 March 20) 90d (666 December 20; fullmoon; major lunar standstill)36236d (666 April 28; heliacal rise of Venus; full moon)8d(inferior conjunction of Venus)250d (665 August 13) + 260d (666 April 30).


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[4] The characteristics of the Initial Series 9.8.16.16.0, 1 Ajaw 18 Wo, suggest at least othersolar sequence based on anomalous multiples, as follows: 9.8.16.16.0 (610 April 18) +9100d (635 March 18) + 33280d (726 April 30) + 68900d (914 December 20) + 68900d(1103 August 12) + 33280d (1194 September 23).

[5] This last sequence might be linked to the mythical record of Palenque 1.18.5.4.0:3710.18.9.15.0, 1 Ajaw 18 Kayab (1194 September 23)1366560d (2548 B.C. March 19) +68900d = 1.18.5.4.0, 1 Ajaw 13 Mak (Birth of GII, on 2360 B.C. November 8, and solarnadir38 located 33280 days before the 819-day station and winter solstice of 2269 B.C.).

[6] The accession date of Pakal on 9.9.2.4.8, 5 Lamat 1 Mol (615 July 29), 39 might beprojected in solar terms, as follows: [6.1] 9.9.2.4.8185120d (108 September 24)33280d(17 August 12); [6.2] 9.9.2.4.8 + 68900d (804 March 19) + 185120d + 9100d (1335December 21) + 68900d (1524 August 12).

[7] When calculating the interval from A.D. 17 August 12 to A.D. 1524 August 12, aseparation of 550420 days (= 1508 x 365d = 1507 x 365.2422d) is obtained, which can beformulated based solely on the anomalous multiples of the Venus Table, as follows: (9100d+ 33280d + 2 x 68900d + 2 x 185120d), which is a significant equivalence.

[8] The Dresden Codex Serpent Series40 8.16.3.12.3, 13 Ak'bal 11 Yaxk'in, and 10.6.10.6.3,13 Ak'bal 1 K'ank'in, are separated by (33280d + 185120d), the solar nadir of A.D. 451October 31 being their common linking point, since 8.16.3.12.3 (pp. 31, 63 of the Dresden)+ 33280d (451 October 31) + 185120d = 10.6.10.6.3 (pp. 31, 62, 63 of the Dresden).

[9] The base date 9 Kan 12 Kayab, for calculation of the Dresden Codex Serpent Series, islocated 10967536 days before the Era Date.41 When applying a peculiar 33280-day intervalto 9 Kan 12 Kayab, the Gregorian date 33051 B.C. August 22 results. The Gregoriancalendar accumulates one day of error for each 3333.33 years, so it is required to project thedate obtained towards a region where our calendar is still effective. Such a projection isachieved through an interval of 11 x 29 x 37960d = 33176 x 365d (i.e. 33154 x 365.2422d),resulting in the Gregorian date of A.D. 104 August 12. Hence, the date 9 Kan 12 Kayabprecedes by 33280 days, the zenith passage of the sun of 33051 B.C., in deep time.

[10] The projection of 40 x 37960d from the accession of Pakal,42 towards A.D. 4772October 23 (1.0.0.0.0.8, 5 Lamat 1 Mol) precedes the Gregorian date of A.D. 4797September 22 by 9100 days. Going back 29 x 37960d, to regions where our calendar is stilleffective, leads to A.D. 1783 September 23. Consequently, the date located 9100 days after1.0.0.0.0.8, corresponds to the autumnal equinox of A.D. 4797, in the distant future.

A complete summary of other potential solar intervals is presented in Appendix I.

Acknowledgments

I am grateful to Vctor Torres Roldn, Ivan prajc, Michael Grofe, Susan Milbrath, EdBarnhart, and Sid Hollander for having commented on this document on a private basis.


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REFERENCES

1. H. Bricker, V. Bricker,Astronomy in the Maya codices (Philadelphia, 2011).2. C. P. Bowditch, The numeration, calendar systems and astronomical knowledge of the Mayas

(Cambridge, 1910).3. J. Teeple, Maya astronomy, Contributions to American archaeology (vol. 1, Washington,

DC, 1930), 38-40, 71-76.

4. M. Len-Portilla, Time and reality in the thought of the Maya (Oklahoma, ed. 2, 1988).5. J. Teeple, op. cit. (ref. 3).6. J. Teeple, op. cit. (ref. 3).7. R. J. Sharer, L. P. Traxler, The ancient Maya (Stanford, ed. 6, 2006), 320-21.8. R. J. Sharer, L. P. Traxler, op. cit. (ref. 7).9. A. Aveni, Skywatchers (Austin, TX, 2001).10.Z. Nuttal, Nouvelles lumires sur les civilizations Americaines et le systme du calendrier,

Proceedings of the 22ndInternational Congress of Americanists (Rome, 1928), 119-48.

11.O. Apenes, Possible derivation of 260 day period of the Maya calendar (vol. 1, Stockholm,1936), 5-8.

12.V. Malmstrm, Origin of the Mesoamerican 260-day calendar, Science 181 (1973); publishedonline 7 September 1973.

13.S. Milbrath, Star gods and astronomy of the Aztecs, La antropologa americanista en laactualidad: homenaje a Raphael Girard(vol. 1, Mxico, 1980), 289-303.

14.J. E. S. Thompson, A commentary of the Dresden Codex, a Maya hieroglyphic book(Philadelphia, 1972).

15.E. Frstemann, Commentary on the Maya manuscript in the Royal Public Library of Dresden ,Papers of the Peabody Museum of American Archaeology and Ethnology, Harvard Univ. (vol.4, no. 2, Cambridge, MA, 1906).

16.F. Lounsbury, A Derivation of the Mayan-to-Julian calendar correlation from the DresdenCodex Venus chronology, The sky in Maya literature, A. Aveni, Ed. (New York, 1992).

17.S. Milbrath, Star gods of the Maya: astronomy in art, folklore and calendars (Austin, TX,1999), 163-177.

18.J. Teeple, op. cit. (ref. 3), 94-98, 110-14.19.C. Barrera,Dos posibles soluciones para el intervalo de 9100 das de las Tablas de Venus del

Cdice de Dresde (Bogot, 2007).

20.F. Lounsbury, Maya numeration, computation, and calendrical astronomy, Dictionary ofscientific biography, vol. 15 (suppl. 1, New York, 1978), 759-818.

21.J. E. S. Thompson, op. cit. (ref. 14).22.F. Lounsbury, op. cit. (ref. 20).23.J. E. S. Thompson, op. cit. (ref. 14).


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24.J. E. S. Thompson,A correlation of the Mayan and European calendars, Anthropological series(vol. 17, no. 1, Chicago, 1927).

25.F. Lounsbury, op. cit. (ref. 16).26.B. Wells, The Venus Table of the Dresden Codex and the movements of the planet Venus,

The Journal of the Royal Astronomical Society of Canada (vol. 85, no. 6, 1991).

27.C. Barrera, Desarrollo de estructuras Venus-solares en ambientes cronolgicamente puroslibres de los efectos correlativos (Bogot, 2012).

28.C. Barrera, op. cit. (ref. 27).29.M. Paxton, Iconografa solar en la Tabla de Venus del Cdice de Dresde, Yucatn a travs de

los siglos: memorias del 49 Congreso Internacional de Americanistas (Quito, 1997), 95-120.

30.F. Lounsbury, op. cit. (ref. 16).31.M. Grofe, Measuring deep time: the sidereal year and the tropical year in Maya inscriptions,

Proceedings of the International Astronomical Union Symposium No. 278 (Per, 2011), 214-30.

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hieroglyphic text of the Temple of the Cross at Palenque, M. Greene, Ed. (Austin, TX, 1980),99-115.

35.C. Barrera, Trnsitos de Venus en el Cdice de Dresde (Bogot, 2009).36.A. Mndez, E. Barnhart, C. Powell, C. Karasik. Astronomical observations from the Temple of

the Sun,Archaeoastronomy (vol. 19, 2005), 44-73.

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and astronumerology (Greensboro, 2008).

39.L. Schele, P. Mathews, The code of kings (New York, ed. 1, 1999).40.H. Beyer, Emendations of the Serpent Numbers of the Dresden Maya Codex, Anthropos

(vol. 28, 1933), 1-7.

41.H. Beyer, The Long Count position of the Serpent Number dates, Proceedings of the 27thInternational Congress of Americanists (vol. 1, Mexico, 1939), 401-05.

42.L. Schele, P. Mathews, op. cit. (ref. 39).


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APPENDIX IPOTENTIAL SOLAR INTERVALS BASED ON MAYA CYCLES

Solar Equations Based on the Anomalous Multiples of the Dresden Codex Venus Table

1. June 21 + 68900d = February 10/112. June 2168900d = October 29/303. June 21 + 3 x 33280d = October 28/294. June 203 x 33280d = February 11/125. December 21 + 68900d = August 12/136. December 2168900d = April 30/May 17. December 21(33280d + 68900d) = March 19/208. December 21 + (9100d + 185120d) = September 22/239. December 21(9100d + 185120d) = March 19/2010.December 21 + 3 x 33280d = April 29/3011.December 21 + (68900d33280d185120d2 x 9100d) = October 28/2912.December 203 x 33280d = August 13/1413.December 20 + (33280d + 68900d) = September 2314.March 19/20 + 33280d = April 30/May 115.March 19 + (33280d + 2 x 68900d) = August 1316.March 19 + (33280d + 68900d + 185120d + 9100d) = September 22/2317.March 19 + 2 x (33280d +68900d) = September 2318.September 2333280d = August 12/1319.October 29/30 + (9100d + 33280d68900d) = March 20/2120.February 11/12(9100d + 33280d68900d) = September 22/2321.April 30 + (33280d + 2 x 68900d) = September 2322. (9100d + 33280d + 2 x 68900d + 2 x 185120d) = 1507 solar years = 1508 x 365d23.(68900d + 3 x 33280 d) = 462 solar years1.9d24.(2 x 68900d + 4 x 33280d185120d9100d) = 210 solar years0.9d25.(185120d + 9100d33280d68900d) = 252 solar years1d26.March 10 + 68900d = October 3127.November 8 + 33280d = December 2128.November 868900d = March 1929.November 8 + (33280d68900d) = April 30/May 1

Solar Equations Based on the 37960-day Cycle

1. December 20/212 x 37960d = February 9/102. December 20 + 2 x 37960d = October 30/313. April 30/May 12 x 37960d = June 204.

April 28 + 3 x 37960

d

= February 11/125. August 133 x 37960d = October 286. February 9 + 4 x 37960d = October 317. March 19/20 + 7 x 37960d = September 22/238. March 19/2013 x 37960d = February 9/109. October 28/2913 x 37960d = September 22/2310.September 2216 x 37960d = October 3111.September 22/2336 x 37960d = March 19/20


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12.November 25 + 37960d = October 3113.November 2537960d = December 21

Solar Equations Based on the 18980-day Calendar Round

1. March 19/20 + 3 x 18980d = February 10/112. March 19 + 11 x 18980d = October 313. December 20/21 + 25 x 18980d = February 9/104. December 2025 x 18980d = October 315. April 30/May 1 + 25 x 18980d = June 20/216. October 30/31 + 3 x 18980d = September 22/237. February 9 + 11 x 18980d = September 23

Solar Equations Based on 2340-day Multiples

1. March 18 + 2340d = August 132. April 28 + 2340d = September 233. August 11 + 2340d = September 234. February 9 + 3 x 2340d = April 305. August 12 + 3 x 2340d = October 31


1. November 82 x 32760d = June 202. March 103 x 32760d = February 9/10


1. June 21 + 11960d = March 202. June 2111960d = September 22Solar Equations Based on 819-day Multiples

1. September 23 + 819d = December 212. December 21 + 819d = March 193. February 9 + 18 x 819d = June 214. June 20/21 + 18 x 819d = October 31/November 1


1. August 12/13 + 260d = April 30/May 12. November 25 + 260d = August 12/133. 1999 x 260d = 1423 solar years4. 2235 x 260d = 1591 solar years5. * 2117 x 260d = 1507 solar years6. 118 x 260d = 84 solar years = 83 Uranus cycles


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1. October 29/30 + 234d = June 20/212. June 20/21 + 234d = February 9/103. April 30/May 1 + 234d = December 20/214. December 21 + 234d = August 12/135. September 22/23 + 585d = April 30/May 16. August 12/13 + 585d = March 19/207. April 19 + 585d = November 25

Solar Equations Based on K'atun Periods

1. August 13 + 1 K'atun = April 30/May 12. August 13 + 4 K'atuns = June 20/213. August 121 K'atun = November 254. August 132 K'atuns = March 10

Solar Equations Based on the Anomalous Multiples and 2340-day Multiples

1. September 22/23 + (9100d2 x 2340d) = October 30/312. February 9 + (9100d2340d) = August 133. March 20 + (9100d2 x 2340d) = September 224. December 20 + (185120d + 2340d) = March 205. December 21 + (9100d + 33280d + 2 x 68900d2 x 2340d) = June 216. October 30 + (9100d + 33280d + 2 x 68900d2 x 2340d) = April 30

Solar Equations Based on the Anomalous Multiples and the 11960-day Cycle

1. December 21(33280d + 11960d) = February 9/102. February 9/10 + (11960d + 33280d + 68900d) = August 12/133. February 9/10 + (11960d + 33280d68900d) = April 30/ May 14. February 10/11 + (11960d68900d) = March 19/20

Solar Equations Based on the Anomalous Multiples and the 18980-day Calendar Round

1. March 19 + (185120d + 68900d18980d) = September 232. December 20/21 + (185120d33280d18980d) = September 23

Solar Equations Based on the Anomalous Multiples and the 32760day Cycle

1. June 20/21 + (2 x 32760d + 33280d) = December 21/222. June 20/21 + (2 x 32760d68900d) = March 19/203. June 20/21 + (2 x 32760d + 33280d68900d) = April 30/May 1

Solar Equations Based on the Anomalous Multiples, the 37960-day Cycle, and the 26280day Cycle

1. (37960d + 33280d + 26280d) = 267 solar years


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Solar Equations Based on the Anomalous Multiples, the 18980-day Calendar Round, and

the 2340day Interval

1. December 20 + (68900d18980d2 x 2340d) = October 30/312. March 19/20 + (185120d + 68900d18980d + 9100d2 x 2340d) = October 30/313. December 21 + (185120d33280d18980d + 9100d2 x 2340d) = October 30/31

Solar Equations Based on the Uranus Cycle

1. * (5 x 2340d + 18980d) = 84 solar years = 83 Uranus cycles2. * (33280d260d2340d) = 84 solar years = 83 Uranus cycles3. * (9100d + 33280d5 x 2340d) = 84 solar years = 83 Uranus cycles4. (33280d + 9100d + 18980d) = 168 solar years = 166 Uranus cycles5. * (68900d + 3 x 33280d) = 462 solar years1.9d (based on 7670 days)6. * (2 x 68900d + 4 x 33280d185120d9100d) = 210 solar years0.9d7. (2 x 68900d + 3 x 33280d185120d2 x 9100d18980d) = 42 solar years8. * (185120d + 9100d33280d68900d) = 3 x 30680d = 252 solar years1d9. (2 x 37960d2 x 32760d33280d + 68900d) = 1.5 x 84 years = 126 solar years10.(185120d + 9100d33280d68900d364d) = 251 solar years = 157 Venus cycles11.(185120d + 9100d33280d68900d364d) = 251 solar years = 248 Uranus cycles

Solar Equations Based on Maya SuperNumbers

1. * March 20 + 1366560d = September 232. August 13 + 1364360d = February 93. September 22 + 1364360d = March 20/214. November 25 + 3276 x 365d = September 235. January 153276 x 365d = March 19/20

Solar Equations Based on Venus Sub-Intervals

1. April 28/29 + 236d = December 212. December 21 + 90d = March 203. March 20 + 250d = November 254. April 28/298d250d = August 13

Solar Equations Based on 182 and 180day Intervals

1.

October 29/30 + 182d

= April 29/302. December 21 + 182d = June 213. August 13 + 180d = February 94. September 22 + 180d = March 21

*Equation previously listed

SOLAR INTERVALS OF THE DRESDEN CODEX VENUS TABLEBY CARLOS BARRERA ATUESTA, BOGOT, D.C., COLOMBIA.

Documents

Official Article on Solar Intervals